equation is limited to LC oscillators as it is dependent on quality factor. In this paper, we present the Generalized Adler's equation applicable for injection locking ...
5C-5 Gen-Adler: The Generalized Adler’s Equation for Injection Locking Analysis in Oscillators Prateek Bhansali
Jaijeet Roychowdhury
ECE Department University of Minnesota Minneapolis, MN, 55455 e-mail: bhansali@umn.edu
ECE Department University of Minnesota Minneapolis, MN, 55455 e-mail: jr@umn.edu
Abstract— Injection locking analysis based on classical Adler’s equation is limited to LC oscillators as it is dependent on quality factor. In this paper, we present the Generalized Adler’s equation applicable for injection locking analysis on oscillators independent of the circuit topology. The equation is obtained by averaging the PPV phase macromodel. The procedure is considerably simple and handy to determine the locking range for arbitrary shape small AC injection signal. Analytical equations for injection locking dynamics are formulated using the Generalized Adler’s equation and validated with the PPV simulations.
I. I NTRODUCTION Injection locking is a nonlinear phenomenon observed in oscillators. A small AC injection signal of frequency within the “locking range” entrains the oscillator at its frequency. The oscillator is thus said to be locked and the phenomenon is known as injection locking. Injection locking is exploited in the frequency synthesis and related applications. For example, it is used in the design of high performance quadrature oscillators [1], [2] and injection locked phase-locked loops (PLLs) [3]. Injection locking also has wide applications in optics [4], [5]. Thus, apart from its theoretical importance, study of injection locking is of great practical interest. Injection locking has been widely studied for LC-tank based oscillators [6] using the classical Adler’s equation [7]. Adler’s equation provides quick insight into the locking dynamics. Furthermore, it provides rapid estimation of the locking range of oscillators for a given injection signal. The locking range is the frequency range around the oscillator’s natural frequency for which injection locking occurs. But, Adler’s equation is dependent on quality factor Q, which limits its use to LC oscillators. The Q factor by its definition is the ratio of energy preserved to energy dissipated per cycle in the oscillator. In LC oscillators, energy is stored in inductors and capacitors and a part of it is dissipated across the resistance. In a ring oscillator this definition does not hold good as there is no energy storage. There are no inductors in a ring oscillator and capacitances are charged and discharged within each oscillator cycle. Recently, analytical techniques to predict injection locking range for ring oscillators have been presented in [8] and numerical techniques in [9] based on the Perturbation Projection Vector (PPV) [10]. These analytical and numerical techniques to obtain the locking range are accurate, being derived mathematically with no assumption of the circuit topology. But, they lack simplicity of Adler’s equation which gives quick and handy graphical insight into the injection locking phenomenon. The alternate method is to study the injection locking using transient simulations. The transient analysis is time consuming as smaller time-steps are required for the accurate simulation of oscillators. The PPV based simulations are already established to be accurate and orders of magnitude faster over transient simulations in predicting injection locking [9]. Furthermore, it takes simulation for hundreds of cycles to conclusively identify the lock condition from the quasi-lock [6]. 978-1-4244-2749-9/09/$25.00 ©2009 IEEE
In this paper, we derive the Generalized Adler’s (Gen-Adler’s) equation from the PPV nonlinear phase macromodel. We apply an averaging technique on the PPV, to average out “fast” varying behavior and retain “slow” varying behavior, obtaining Gen-Adler’s. Based on the analytical PPV of LC-tank oscillator [11], we illustrate the procedure to obtain the classical Adler’s equation as a special case of Gen-Adler’s equation. For an ideal ring oscillator, we formulate Gen-Adler’s equation for sinusoidal, square and exponentially rising and falling AC injection signals based on its analytical PPV [12], giving useful insight into the injection locking phenomenon. The analytical formulation is validated against numerical computations and the locking dynamics are validated against already established PPV methods. It is worthwhile to emphasize that Gen-Adler’s equation can be obtained for any oscillator with arbitrary AC injection signal of small amplitude numerically, but it is an approximation of the PPV. This paper is organized as follows. In Section II, we briefly review Adler’s equation and the PPV phase macromodel. Next, in Section III, we derive Gen-Adler’s equation from the PPV by averaging it. In Section IV, we apply Gen-Adler’s equation to analyze injection locking phase dynamics for both the LC tank oscillator and the ideal ring oscillator giving analytical expressions of the locking range. In Section V, we validate the formulation of Gen-Adler’s and compare results with the PPV based simulations. II. BACKGROUND In this section, we briefly review the classical Adler’s equation [7] and the nonlinear PPV phase macromodel [10] for oscillators. Adler’s equation is the analytical method proposed by Adler to predict injection locking. On the other hand, the PPV is a non-linear phase macromodel for oscillators, suitable for fast and accurate simulation of non-linear phenomenons in oscillators. We use the PPV to derive insightful Adler-like formulations for injection locking in oscillators. A. Adler’s Equation In 1946, R. Adler obtained a differential equation known as Adler’s equation for the oscillator’s phase difference with the injection signal. This celebrated equation describes injection locking dynamics in LC oscillators. When a free running oscillator is perturbed by an AC injection signal, the lock dynamics are given by the following equation Vi f0 dΔφ (t) = Δ f0 − sin(Δφ (t)) dt V 2Q
(1)
where, Δφ (t) is the phase difference. The amplitude of the injected signal is Vi and the output amplitude of oscillator is V while the oscillator runs at a free running frequency of f0 . The frequency difference between the injected signal and the oscillator is Δ f0 . Adler obtained injection locking behavior and the locking range for LC oscillators based on (1). When the oscillator is locked, the phase difference becomes constant and (1) gives 1 Vi Δ f0 sin(Δφ (t)) = f0 2Q V
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(2)
5C-5 Next, differentiating (10) and substituting the value of α˙ (t) from (9), we get
and since sin(·) lies between +1 and −1, we have Δf 1 Vi 1 Vi ≤ 0 ≤ 2Q V f0 2Q V This immediately gives the locking range fL as −
fL = 2|Δ f0 |max f0 Vi fL = QV
(3)
(4)
Next, we review the PPV phase macromodel for oscillators which we will use to derive Adler-like equation. We start with the mathematical equation describing oscillator and list the phase deviation equation (7) under the effect of a perturbation signal [10]. B. The PPV phase macromodel Let the oscillator has a state space x(t). The differential equation system describing it, under the effect of perturbation signal b(t), can be written as dq(x(t)) + f (x) = b(t) (5) dt where, q(·) and f (·) are nonlinear functions. If the perturbation signal is small then the amplitude variations can be neglected, and the solution x p (t) of (5) will be x p (t) =xs (t + α (t))
χ ( f0 (t + α (t))) ·b( f1t) =
dΔφ (t) = −( f1 − f0 ) + f0χ ((Δφ (t) + φ1 (t))) ·b(φ1 (t)) dt
where, vT1 is T-periodic and is called the PPV of the oscillator. The PPV has the same size as x(t). The elements of the PPV gives the phase sensitivity of the corresponding components of x(t) to an externally applied perturbation signal. The PPV can be calculated for oscillators numerically in the time and the frequency domain [13]. Interestingly, the analytical PPV for the LC-tank oscillator and the ideal ring oscillator exist in the literature [11], [12]. III. G ENERALIZED A DLER ’ S E QUATION In this section, we use the PPV phase macromodel reviewed in the last section to obtain Generalized Adler’s equation. We first change the variables in (7) from phase deviation α (t) to phase difference Δφ (t) to obtain a modified phase equation. We then perform averaging in the phase equation retaining the slow behavior to derive Gen-Adler’s. A. Modified Phase Equation In this section, we derive the modified phase equation. Let vT1 (t) be a PPV, periodic with frequency f0 . We can write vT1 (t) = χ ( f0t)
(8)
where, χ (·) is 1-periodic function i.e. χ ( f0t) = χ ( f0t + 1). Assuming the perturbation signal is periodic with frequency f1 , the PPV equation (7) is of the following form d α (t) = χ ( f0 (t + α (t))) ·b( f1t) (9) dt where, b(·) is 1-periodic function now. We can define phase difference between the perturbation signal and the oscillator as Δφ (t) = φ (t) − φ1 (t), for φ (t) = f0 (t + α (t)) and φ1 (t) = f1 t, to obtain Δφ (t) = f0 (t + α (t)) − f1t Δφ (t) f1 − f0 + t α (t) = f0 f0
(10)
(12)
In the modified phase equation (12), Δφ (t) is phase difference between the oscillator and the perturbation signal. For the injection locking analysis, perturbation signal is an AC injection signal with frequency close to the oscillator’s natural frequency.
B. Gen-Adler’s Equation We now assume that in (12), φ1 (t) is “fast” varying and Δφ (t) “slowly” varying variable. This assumption is explained below in detail. Under this assumption, we can average out the fast φ1 (t) and retain the slow Δφ (t) variations in (12) to obtain g(Δφ (t)) as g(Δφ (t)) =
(6)
(7)
(11)
Substituting the value of α (t) from (10) in (11), we obtain a modified phase equation suitable for further analysis
1 T1
T1
χ (Δφ (t) + φ1 (t)) · b(φ1 (t)) d φ1 (t)
0
(13)
where, T1 = φ1 ( f11 ). Hence,
where, xs (t) is the steady state T-periodic solution without any perturbation signal. The phase deviation or the time-shift in steady state solution , α (t), is given by a scalar equation [10] d α (t) =vT1 (t + α (t)) ·b(t) dt
1 dΔφ (t) f1 − f0 + f0 dt f0
g(Δφ (t)) =
1 1
1 0
χ (Δφ (t) + φ1 (t)) · b(φ1 (t)) d φ1 (t)
(14)
Therefore, after averaging out the fast variations, (12) can be written as dΔφ (t) (15) = −( f1 − f0 ) + f0 g(Δφ (t)) dt This is the Generalized Adler’s equation valid for any oscillator as compared to the original Adler’s equation (1), which is only applicable to LC-tank like oscillators. This is of same form as original Adler’s equation for sinusoidal g(Δφ (t)). Under lock conditions, the phase difference between oscillator and injected signal becomes constant, that is dΔφ (t)/dt = 0 or Δφ (t) = Δφ0 and hence f1 − f0 = f0 g(Δφ0 ) Δ f0 = f0 g(Δφ0 )
(16)
Let fL be the locking range of the oscillator. The maximum value of g(·) gives the locking range of the oscillator about f0 as |Δ f0 |max = f0 [g(Δφ (t))]max fL = 2|Δ f0 |max
(17)
The locking range, fL , is typically much smaller than f1 or f0 . If the oscillator is not locked Δφ˙ (t) = 0, and the maximum value of RHS in (15) is −( f1 − f0 ) + fL /2 f1 . Thus, we have dΔφ (t) = −( f1 − f0 ) + fL /2 dt max (18) d φ1 (t) = f1 dt As f1 is close to free running frequency f0 of the oscillator and fL is small, d φ (t) dΔφ (t) 1 (19) dt dt max Hence, our assumption that φ1 (t) is fast varying and Δφ (t) is slowly varying is justified.
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5C-5 IV. I NJECTION LOCKING R ANGE USING G EN -A DLER ’ S E QUATION In this section, we apply the proposed equation in Section III on two oscillators, the LC oscillator and the ring oscillator. We obtain the analytical formulation of Gen-Adler’s for both oscillators. To show the general applicability, we analyze the ring oscillator for various types of injection signals.
assumed to be equal and inverters have ideal characteristics of the following form A if V < 0 (24) f (V ) = −A if V ≥ 0
A. LC Oscillator Consider a simple −Gm LC tank oscillator as shown in Fig. 1. The state variables of this oscillator are voltage across the capacitor v(t) and current through the inductor i(t). The analytical PPV of the LC oscillator [11], when perturbed by a sinusoidal injection signal b(t) = Ii cos(2π f1t) as shown in Fig. 1, is given as vT1 (t) = −
(20)
v(t)
R
V1
R
V2
f(V1)
V3
R f(V2)
C
Fig. 2.
L1 sin(2π f0t) CA
for a steady state solution of v(t) = A cos(2π f0t). i(t)
f(V3)
C
b(t)
C
Ring oscillator circuit with injection signal b(t).
In [12], an analytical PPV for the node V 3 of ideal ring oscillator was derived as ⎧ t ⎨ √1 R e τ if 0 ≤ t < T2 5 A T t v1 (t) = R 2 (25) ⎩ A √ − 1 e τ if T2 ≤ t < T 5
L
Fig. 1.
C
R
where, τ is the RC time constant and T = 2.88727τ = 1/ f0 is time period of the ring oscillator. To obtain the 1-periodic form of the PPV at node V3 we put t = f0 t, which gives us ⎧ t ⎨ √1 R e f0 τ if 0 ≤ t = ft0 < T2 A T 5 t χ (t) = v1 (t/ f0 ) = ⎩ R √2 − 1 e f0 τ if T ≤ t = t < T A 2 f0 5 ⎧ (26) 1 ⎨ √1 R e2.887t if 0 ≤ t < A 2 = R 5 2 ⎩ A √ − 1 e2.887t if 12 ≤ t < 1
f(v) b(t)
LC tank oscillator circuit with injection signal b(t).
5
The PPV can now be written in 1-periodic form using vT1 (t) = χ ( f0 t) as t L1 sin(2π t) χ (t) = vT1 ( ) = − f0 CA (21) L1 sin(2π (Δφ (t) + φ1 (t))) χ (Δφ (t) + φ1 (t)) = − CA To evaluate g(Δφ (t)), we use χ (·) from (21) in (14) and evaluate the integral 1 L1 g(Δφ (t)) = sin(2π (Δφ (t) + φ1 (t))) ·b(φ1 (t)) d φ1 (t) − C A 0 1 L1 sin(2π (Δφ + φ1 )) · Ii cos(φ1 ) d φ1 − = C A 0 Ii L 1 sin(2π Δφ ) =− 2 CA (22) We note that the amplitude of the current through the inductor is given as IL = A CL and current through resistor as IR = QIL . Using this and plugging g(Δφ ) in (15), we obtain the original Adler equation for the LC oscillator as Ii f0 dΔφ = −( f1 − f0 ) − sin(2π Δφ ) (23) dt IR 2Q This is a special case of Gen-Adler’s equation, with a LC oscillator perturbed by a sinusoidal injection signal. As we show next, we can apply Gen-Adler’s equation to study injection locking in ring oscillators directly unlike the original Adler’s equation.
For notational simplicity, K1 RA eK0 ts χ (ts ) = K2 RA eK0 ts
if 0 ≤ ts < 12 if 12 ≤ ts < 1
(27)
where,
1 2 (28) K1 = √ , K2 = √ − 1 5 5 Gen-Adler’s equation for the ring oscillator is then obtained by evaluating g(Δφ (t) from (14) for each of the injection signals. After evaluating g(Δφ (t), we plot it for R = 2kΩ, C = 2nF and A = 1V in each case. The lock range can be easily obtained from these graphs or by finding the maximum of g(Δφ (t). The injection current amplitude was taken to be Ii = 0.1mA and the frequency, f1 = 1.02 f0 , where f0 is free running frequency of the ring oscillator. 1) Sinusoidal Injection Current: For a sinusoidal injection current signal b(t) = Ii sin(2π f1 t) to the ring oscillator shown in Fig. 2, we have K0 = 2.887,
b(φ1 (t)) = Ii sin(2πφ1 (t))
if 0 ≤ φ1 < 1
(29)
χ (Δφ (t) + φ1 (t)) · b(φ1 ) d φ1
(30)
where φ1 (t) = f1 t. Therefore, from (14), (27), and (29) g(Δφ (t)) =
1 0
So, g(Δφ ) for the ring oscillator with sinusoidal injection signal is obtained as g(Δφ (t)) =
B. Ring Oscillator In this subsection, we derive Gen-Adler’s equation for the ring oscillator circuit shown in Fig. 2. All resistances and capacitors are
1 4π 2 + K02
RIi sin(2π Δφ (t) + ζ ) A
× K1 (eK0 /2 + 1) − K2 (eK0 + eK0 /2 )
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(31)
5C-5 where,
or in its 1-periodic form we have Ii if 0 ≤ φ1 (t) < η b(φ1 (t)) = 0 if η ≤ φ1 (t) < 1
2π sin(ζ ) = 4π 2 + K02
and Gen-Adler’s equation for the ring oscillator with sinusoidal injection is dΔφ = −( f1 − f0 ) + f0 g(Δφ (t)) dt The phase lock would occur, when
dΔφ (t) dt
(32)
is 0 i.e.
1 RIi ( f1 − f0 ) sin(2π Δφ (t) + ζ ) = f0 4π 2 + K02 A
× K1 (eK0 /2 + 1) − K2 (eK0 + eK0 /2 )
(33)
and the locking range is given at the maximum value of g(Δφ (t)). In this case, the maximum value of g(Δφ (t)) occurs when sin(2π Δφ (t) + ζ ) = 1, therefore f RIi K1 (eK0 /2 + 1) − K2 (eK0 + eK0 /2 ) |Δ f0 |max = 0 4π 2 + K02 A (34) RIi A and the locking range is fL = 2|Δ f0 |max . This can be easily seen from the Fig. 3 graphically. = 0.6773 f0
The steady state phase difference, Δφ0 , between the oscillator and the injected signal is given by “stable” solution of (33). For Δ f0 / f0 = 0.02, Fig. 3 shows the plot of (33). It can be clearly seen from Fig. 3 that oscillator will lock to the injection signal only if Δ f0 / f0 line intersects g(Δφ (t)), that is (33) has a valid solution. The line Δ f0 / f0 intersects g(Δφ (t)) twice in a period. The two intersection points are marked as U (unstable) and S (stable). When Δφ (t) is perturbed slightly from the unstable point, it will move away from it towards a stable point, depending upon the sign of dΔφ (t)/dt as shown in Fig. 3. For given g(Δφ (t)), the locking range is the maximum range of injection signal frequency, f1 , yielding a valid solution of (33).
(36)
where, φ1 (t) = f1 t. Without loss of generality, we choose 0 < η ≤ 0.5 and evaluate g(Δφ (t)) as
⎧ RIi K1 K0 Δφ (t) (eη K0 − 1) ⎪ e ⎪ A K 0 ⎪ ⎪ ⎪ ⎪ if 0 ≤ Δφ (t) < 12 − η ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎪ ⎪ ⎪ RIi 1 K0 /2 + eK0 Δφ (t) (K eη K0 − K ) ⎪ − K )e (K ⎪ 1 2 2 1 A K0 ⎪ ⎪ ⎪ ⎪ if 12 − η ≤ Δφ (t) < 12 ⎪ ⎪ ⎪ ⎨
g(Δφ (t)) = RI i K2 ⎪ eK0 Δφ (t) (eη K0 − 1) ⎪ A K0 ⎪ ⎪ ⎪ ⎪ ⎪ if 12 ≤ Δφ (t) < 1 − η ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎪ ⎪ RIi 1 ⎪ ⎪ (K1 − K2 )eK0 /2 + eK0 Δφ (t) (K2 − K1 e(η −1)K0 ) ⎪ A K0 ⎪ ⎪ ⎪ ⎪ if 1 − η ≤ Δφ (t) < 1 ⎪ ⎪ ⎩ (37) Similarly, g(Δφ (t)) can be evaluated for 0.5 < η < 1. To evaluate the locking range, we observe that g(Δφ (t)) is composed of four piecewise exponential functions and is symmetric about zero. It is monotonically increasing for 0 ≤ Δφ (t) < 12 − η and decreasing for 1 1 2 − η ≤ Δφ (t) < 2 . Hence, the maximum value of g(Δφ (t)) should occur at φ (t) = 1/2 − η . Thus, the injection locking range for the ring oscillator with square wave injection signal is gives as RIi K1 K0 /2 fL = 2 f0 (1 − e−η K0 ) e (38) A K0 Fig. 4 shows the plot of (37) with the unstable and stable point for one period.
0.15 g(Δφ(t)) Δf /f 0 0
0.1
dΔφ(t)/dt>0
(Δf /f ) 0 0 max
0.15
0 0
g(Δφ(t), Δf /f
g(Δφ(t)) Δf0/f0
g(Δφ(t), Δf0/f0
(Δf /f ) 0 0 max 0.1
dΔφ(t)/dt>0
0.05
S
U
0.05
S
U 0
−0.05
0
dΔφ(t)/dt
